Exact concentration regions for stimulus-switched competitive binding under finite depletion

Abstract

Transferring binary binding constants into competitive mixtures is a common but rarely validated step in stimulus-programmed molecular recognition. A pH-programmed Co8L12 coordination cage reported by Cullen et al., Chem. Sci., 2015, 6, 4025–4028 supplies a parameter-free test because its reported C/B composition at pH 7.5 requires KeffB(7.5) < 6.39 M−1, yet the entire rectangle formed from the quoted marginal affinity ranges lies above this bound, with the most favourable point still giving 8.78 M−1. This study develops a method that accounts for finite depletion, which separates concentration-window limitations from parameter-transfer failure. For two guests with one total fixed, monotonicity converts independent fidelity and occupancy targets into one-sided thresholds whose intersection is the complete admissible interval. A pairwise endpoint bound extends the consistency test to mixtures with additional same-site competitors, and a free-host-parametrized boundary construction maps the admissible region when two totals vary independently. For the Co8L12 cage reported by Cullen et al., Chem. Sci., 2015, 6, 4025–4028, a reference model neglecting depletion predicts a controlled-free-concentration reference of 95.95%, while the exact finite-depletion maximin is restricted to 94.36%. Conversely, a pH-responsive naphthotube operating with a wider selectivity margin retains a broad 95% target window and requires only a minor 0.22 percentage-point adjustment for depletion effects. The method therefore identifies when finite depletion changes a design decision and when transferred binary constants are inconsistent with competitive-mixture observations.